You can also choose to pass the FE exam to qualify for the patent bar; most states require a BS in engineering to take the FE exam, but not all, and an FE pass in a state that doesn't still qualifies.
There are also various combinations of coursework that qualify (x semester hours of physics, y semester hours of engineering or chemistry, etc...). It's in the patent exam application docs.
One interesting and controversial one is that mathematics courses are excluded from patent prep (i.e., mathematics is specifically excluded from the list of courses that qualify you for the patent bar[1]).
[1] perhaps this is why mathematics dressed up as something else often gets patented, because the patent bar specifically and deliberately excludes people with strong math backgrounds from the profession.
Yeah, some of the requirements do not seem well thought out. I've passed the EIT and PE exams but not the FE exam. I have two engineering degrees, but let's say I had a math degree.(If I recall correctly you can take the EIT/FE/PE with hard science or math degrees.) I wouldn't qualify for the patent exam. That said, all these licensing exams are about jumping through hoops- find out the requirements, and finish them.
